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Fundamental Principles of Tetration
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(03/08/2016, 10:18 AM)sheldonison Wrote: Hi Daniel,
Welcome back.
Sheldon, thank you for your gracious welcome.

(03/08/2016, 10:18 AM)sheldonison Wrote: Probably the two biggest ones are that Kneser's solution has been "rediscovered", as the preferred solution for extending tetration to the real and complex numbers.
In what manner is Kneser's solution preferred? Please explain the mathematics behind it or provide links. My understanding is that use of Schroeder's equation and Abel's equation are exclusive. Abel's equation is for limit point where and Schroeder's equation is for

(03/08/2016, 10:18 AM)sheldonison Wrote: This second complex tetration base program generates the Abel function on a sickle, exactly meeting the uniqueness criteria. It works by solving the problem for iterating instead of where , so k=0 (which is parabolic; with a formal asymptotic series for the two solutions) corresponds to base . There is a linear transformation, so the two problems; iterating in y or z, are congruent. This is what they call the perturbed fatou coordinate in complex dynamics, which is equivalent to an slog/abel function starting by perturbing bases .
This looks interesting. I'm late to the game of looking for a tetration that maps the reals into the reals.

(03/08/2016, 10:18 AM)sheldonison Wrote: There are other solutions for tetration as well, Kouznetsov's solution, and Andrew's slog using simultaneous equations, which appear to give the same solution ...
I am very interested in the different solutions for tetration and how they fit together. Aldrovandi uses Bell matrices for both fractional iteration and tetration. My own work on fractional iteration is numerically consistent with Aldrovandi's work. I am somewhat familiar with Gottfried Helm's work with Bell and Carleman matrices.
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Messages In This Thread
Fundamental Principles of Tetration - by Daniel - 03/08/2016, 03:58 AM
RE: Fundamental Principles of Tetration - by Daniel - 03/10/2016, 02:14 AM

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